Prime Difference Function

$$d_n\equiv p_{n+1}-p_n.$$

The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (Sloane's A001223). Rankin has shown that

$$d_n>{c\ln n\ln\ln n\ln\ln\ln\ln n\over(\ln\ln\ln n)^2}$$

for infinitely many $n$ and for some constant $c$ (Guy 1994).

An integer $n$ is called a Jumping Champion if $n$ is the most frequently occurring difference between consecutive primes $n\leq N$ for some $N$ (Odlyzko et al. ).

See also Andrica's Conjecture, Good Prime, Jumping Champion, Pólya Conjecture, Prime Gaps, Shanks' Conjecture, Twin Peaks

References

Bombieri, E. and Davenport, H. ``Small Differences Between Prime Numbers.'' Proc. Roy. Soc. A 293, 1-18, 1966.

Erdös, P.; and Straus, E. G. ``Remarks on the Differences Between Consecutive Primes.'' Elem. Math. 35, 115-118, 1980.

Guy, R. K. ``Gaps between Primes. Twin Primes'' and ``Increasing and Decreasing Gaps.'' §A8 and A11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19-23 and 26-27, 1994.

Odlyzko, A.; Rubinstein, M.; and Wolf, M. ``Jumping Champions.'' http://www.research.att.com/~amo/doc/recent.html.

Riesel, H. ``Difference Between Consecutive Primes.'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, p. 9, 1994.

Sloane, N. J. A. Sequence A001223/M0296 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.